French Matra driver Jean-Pierre Beltoise drifting through trainstation corner [1]
Why Does A Tyre Grip The Road? Part 4.1 - Lateral Grip
This is part 4.1 of our Why Does A Tyre Grip The Road series, today we will be looking into Lateral grip. This post blew out a bit, so I decided to write it in two parts, it has been written, along with the rest of the series assuming you have read the ones before it. But it's not necessary to read all of them, it just helps to provide a bit more colour, read the first one on Rubber here, the second one on Grip here, the third one on Longitudinal Grip here, and the second part on Lateral Grip here.
Lateral Grip
Where Fc = Centrifugal Force
Y = Centripetal force
m = Mass of the vehicle
v = Speed of the vehicle
r = Radius of the bend
The centripetal force's grip equation is similar to the longitudinal grip equation, we can again use Newton’s second law to create it
Where Y = Lateral grip force (Centripetal force)
t= Coefficient of transversal friction
Z = Vertical load
As with the longitudinal equation, this one is just as loaded. But before we get into that, as we have two equations for the centripetal force, let's put them on opposite sides of an equal sign and see if we can find out anything of interest. Right now we are most concerned with cornering speed, so let's look at that. With a little algebraic magic and remembering that the vertical load is just the mass multiplied by the gravitational constant, we can solve for velocity
Where g = Gravitational acceleration
This equation shows us that the speed a tyre can turn a corner, with a radius r, can only be affected by the coefficient of transversal friction, interesting, but not at all unpredictable given how simple we have kept the analysis so far. If we take a step back and look at the whole bend, we can further increase our cornering speed by altering our driving line. If we use all of the road, increasing the radius of the bend, we can impact the cornering velocity just as much as the coefficient of transversal friction. That’s quite interesting and something we will be covering in upcoming posts. But for now, let's look into what makes up the coefficient of transverse friction.
The above equation assumes that there is no aero on the car, but what if there is, what happens then? Well things get a little more complicated, but nothing too extreme, we simply add the down force generated by the aero into the above equation, which now looks like this
Where L = downforce
What's interesting about this equation is that unlike the one in the body of text, the mass of the vehicle directly impacts the cornering speed. Reducing the mass increases the effect of the downforce and through this increases the cornering velocity. Given that there is a limit to how much weight can be removed from a car, either from regulations, or due to structural issues, our best bet is to increase downforce, which strangely enough, increases with velocity.
Where L = Lift (negative for downforce)
Cl = Coefficient of lift, governed by the shape of wing used and its angle of attack. For anything other than basic shapes it's found experimentally
ρ = density of the air
v = velocity
A = the frontal area of the car
We can therefore go faster around a corner, by going faster around a corner! Obviously this is again a simplification of what is going on, we are looking at just one tyre, we are still limited by τ, and as with all things in race cars, increasing downforce has a cost, it increase drag. There is again a delicate balancing act, one where depending on the type of car you are building and conditions on the day, you get to experiment, you get to have fun.
Slip angle
Now that we have had a bit of a look at what governs the velocity for a given corner, let's take a close look at what is creating the grip, what gives us our τ.
Slip angle of a tyre [2]
Whilst the tyre isn’t pointing in the direction of travel, within the contact patch the rubber deforms to the point where it is pointing in the direction of travel. The level of deformation is progressive, as you can see in the above image, the rubber twists from the pointing direction to the direction of travel, twisting back as it moves out of the contact patch. It’s this that creates the friction between the tyre and the road, giving us grip.
Unfortunately the relationship between slip angle and grip isn’t linear, increasing the slip angle only increases our grip to a point, after which it fades away. The reduction in grip is caused by increasing tyre temperatures, the higher temps push the rubber into its rubbery state and reducing our grip.
Lateral force vs slip angle [3]
The point of maximum grip depends on a number of things, the tyre’s construction, the rubber compound, the road surface, amount of vertical load, the tyre's inflation pressures, to name a few, but typically ranges in between 3 to 7 degrees of slip angle for a race tyre. Below is a great video demonstrating a tyre's slip angle being tested.
Tyre Load Sensitivity
A tyre's level of grip is, as stated above, influenced by a large range of things, but the most important to us at the moment is the effect varying vertical loads have. As vertical loads are increased, so too are the tyre's maximum level of grip. But it's not just the grip level that increases, as you can see below, the amount of lateral force for all slip angles increases (the curve becomes steeper) and the slip angle where peak lateral force is generated has also increased.
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Lateral force vs. slip angle [4]
Unfortunately this relationship isn’t one to one, there is diminishing returns, the more we increase the vertical load the less lateral force we get for each increase. This is known as the tyre’s load sensitivity, which is essentially the decreasing gain in lateral force from increasing loads. A better way to show this is by plotting the lateral force coefficient against the slip angle.
Where Fy = Lateral force
Fz = Normal force
Below I have created this graph by using the data in the graph titled ‘Lateral force vs vertical load for a range of slip angles’ below. Unfortunately the data only goes up to 6°s of slip angle, but we can still see the impact increasing loads have. As loads are increased we reduce our lateral force coefficient, we are getting a smaller increase in grip with increasing levels of load.
Lateral force coefficient vs slip angle for a range of vertical loads, data from [5]
Clearly vertical loads impact on grip, through the changing slip angle and the tyre’s load sensitivity, are key things to consider when setting up steering and suspension, but these aren’t the only things that impact grip.
Cornering Stiffness
Another important consideration is a tyre’s cornering stiffness, it governs the tyre’s performance under low slip angles and low side forces. Cornering stiffness, despite what the name implies, isn’t talking about the distortion of the tyre, that is its lateral stiffness, we will cover that in the next post. We can see a tyre’s cornering stiffness when we plot lateral force vs slip angle, it is the slope of the curve within the elastic region (linear region), C in the graph below.
Lateral grip force vs slip angle with cornering stiffness highlighted [5]
If we remember back to high school maths, we know that the slope of the curve is rise/run, or in this case
Ιf we rearrange it, we can define lateral force as
A tyre with a higher cornering stiffness creates a higher lateral force for a given slip angle.
This sounds great, but what does it actually mean for us driving around? Well it means that because a lower slip angle generates less temperature, assuming our tyres are in their operating temp, our tyres will last longer, but more importantly, it means we can run a softer compound, giving us even more grip. If we have a look at the graph below, we can see this when we compare the F5000 tyre with the street car tyre. To generate an equal coefficient of transverse friction, the street tyre needs to create a slip angle of 8° whereas the Formula 5000 tyre can do it with just 3°s of slip.
.jpg "Coefficient of friction vs slip angle for a range of tyres")
Coefficient of friction vs slip angle for a range of tyres [6]
From what I have said above, it might seem that a higher cornering stiffness is the way to go, and for the most part, you’d be right, but it’s just not that simple. Changing the cornering stiffness changes the driving characteristics of the car dramatically. If we look at the hypothetical tyre in the graph above, we can see that maximum grip is achieved from low slip angles, but turning the wheel further than the max will make the grip feel like it has fallen off a cliff. You will still have grip at this point, but it won’t be too confidence inspiring. Compare that to the road tyre, where max grip develops slower, and once there, stays fairly consistent for a range of slip angles. If the driver created a slip angle where it was on the downward slope, say 20°, it is still a gradual loss of grip, nothing like the roller coaster shaped hypothetical tyre.
Options for Increasing Cornering Stiffness
But what if you are in a race series where tyres are mandated is there a way to increase cornering stiffness for a given tyre? Yes, you have two options, increase the inflation pressure or increase the vertical load (increasing the rim width for a given tyre will increase it as well). As you can see below, an increase in the tyre pressures increases the cornering stiffness, the increase in air pressure makes the tyre carcass harder to deform and therefore stiffer.
Cornering stiffness vs inflation pressures [5]
Our other option is to increase the vertical load on the tyre, as you can see below, an increase in the vertical load increases the cornering force, or lateral force, up to a point.
Lateral force vs vertical load for a range of slip angles [5]
As you probably expected, you can’t change one thing without affecting the other, changing the vertical load affects what cornering stiffness you get for a tyre pressure and vice versa. Schmeitz, A [7] found that at low vertical loads, a lower inflation pressure had higher cornering stiffness, whilst at higher loads, higher inflation pressure had a higher cornering stiffness, see graph below.
Cornering stiffness vs vertical load for a range of inflation pressures [7]
Schmeitz, A [7] states that there are two contradictory effects going on, the first is related to the fact that for a given vertical load, the contact patch reduces with increasing pressures, a smaller contact patch length reduces the cornering stiffness of a tyre (see drop down for a little info explaining this using the brush model). This effect takes over at the lower vertical loads and explains why lower pressures have a higher cornering stiffness in this region. The other reason explaining why higher pressures dominate at the higher vertical loads is that the increased inflation pressure increases the lateral stiffness, it deforms less from a given lateral force. If we refer to the Brush Model’s equation for lateral force in the drop down below, we can see that a higher lateral stiffness for a given slip angle will result in a higher lateral force. This gives us a steeper lateral force vs slip angle curve and therefore a higher cornering stiffness. Whilst these rules are generally correct, as you can see above they don’t hold for ever increasing, or decreasing pressures and could change completely depending on the tyre. The inflation pressure that had the best cornering stiffness across the whole vertical load range was the middle pressure 2.4bar, highlighting that we need to find the optimum pressure for our tyre.
The Brush model is a simplified tyre model that assumes tyres are made up of bristles, see image below. The complete carcass is allowed to deform towards the spinning axis and the bristles are able to deform independently of one another. The bristles are initially vertical when they enter the contact patch and assuming the tyre is free rolling, they will stay this way. When an outside force is applied, both lateral and longitudinal in the image below, the bristles are deflected in the direction of the force. The deflection of the bristles is a representation of the carcass, belt, and tread elements deforming.
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The simplifying assumptions allows for some simplified equations, the lateral force, aligning torque, and cornering stiffness equations are below
Where Cpy= Lateral Stiffness
a = Half of the contact patch length
α = Slip angle
As you can see above, using this simple but powerful model shows us that the cornering stiffness is directly related to the contact patch length, any reduction in it will lower our cornering stiffness.
Why Does A Tyre Grip The Road Series
References:
[2] Balkwill, J. 2017, Performance Vehicle Dynamics
[3] Michelin, The Tyre Grip, Link, retrieved on 03/03/22
[4] Milliken, W. Milliken, D. 1995, ‘Race Car Vehicle Dynamics’
[5] Vorotovic, G. et al. 2013, 'Determination of Cornering Stiffness Through Integration of A Mathematical Model and Real Vehicle Exploitation Parameters'
[6] Smith, C. 1978, ‘Tune to Win - The art and science of race car development and tuning’
[7] Schmeitz, A. et al. 2005, ‘Extending the Magic Formula and SWIFT tyre models for inflation pressure changes’, Link retrieved on 06/04/22
[8] Pacejka, H. 2012, Tyre and Vehicle Dynamics
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